I PROOFS.- 1 Informal Logic.- 1.1 Introduction.- 1.2 Statements.- 1.3 Relations Between Statements.- 1.4 Valid Arguments.- 1.5 Quantifiers.- 2 Strategies for Proofs.- 2.1 Mathematical Proofs - What They Are and Why We Need Them.- 2.2 Direct Proofs.- 2.3 Proofs by Contrapositive and Contradiction.- 2.4 Cases, and If and Only If.- 2.5 Quantifiers in Theorems.- 2.6 Writing Mathematics.- II Fundamentals.- 3 Sets.- 3.1 Introduction.- 3.2 Sets - Basic Definitions.- 3.3 Set Operations.- 3.4 Indexed Families of Sets.- 4 Functions.- 4.1 Functions.- 4.2 Image and Inverse Image.- 4.3 Composition and Inverse Functions.- 4.4 Injectivity, Surjectivity and Bijectivity.- 4.5 Sets of Functions.- 5 Relations.- 5.1 Relations.- 5.2 Congruence.- 5.3 Equivalence Relations.- 6 Infinite and Finite Sets.- 6.1 Cardinality of Sets.- 6.2 Cardinality of the Number Systems.- 6.3 Mathematical Induction.- 6.4 Recursion.- III Extras.- 7 Selected Topics.- 7.1 Binary Operations.- 7.2 Groups.- 7.3 Homomorphisms and Isomorphisms.- 7.4 Partially Ordered Sets.- 7.5 Lattices.- 7.6 Counting: Products and Sums.- 7.7 Counting: Permutations and Combinations.- 8 Number Systems.- 8.1 Back to the Beginning.- 8.2 The Natural Numbers.- 8.3 Further Properties of the Natural Numbers.- 8.4 The Integers.- 8.5 The Rational Numbers.- 8.6 The Real Numbers and the Complex Numbers.- 8.7 Appendix: Proof of Theorem 8.2.1.- 9 Explorations.- 9.1 Introduction.- 9.2 Greatest Common Divisors.- 9.3 Divisibility Tests.- 9.4 Real-Valued Functions.- 9.5 Iterations of Functions.- 9.6 Fibonacci Numbers and Lucas Numbers.- 9.7 Fuzzy Sets.- Appendix: Properties of Numbers.- Hints for Selected Exercises.- References.

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.